Asymptotic expansions for nonlocal diffusion equations in Lq-norms for 1⩽q⩽2

AbstractWe study the asymptotic behavior for solutions to nonlocal diffusion models of the form ut(x,t)=J∗u(x,t)−u(x,t)=∫RdJ(x−y)u(y,t)dy−u(x,t) in the whole Rd with an initial condition u(x,0)=u0(x). Under suitable hypotheses on J (involving its Fourier transform) and u0, it is proved an expansion of the form‖u(x,t)−∑|α|⩽k(−1)|α|α!(∫u0(x)xαdx)∂αKt‖Lq(Rd)⩽Ct−A, where Kt is the regular part of the fundamental solution and the exponent A depends on J, q, k and the dimension d. Moreover, we can obtain bounds for the difference between the terms in this expansion and the corresponding ones for the expansion of vt(x,t)=−(−Δ)s2v(x,t). Here we deal with the case 1⩽q⩽2. The case 2⩽q⩽∞ was treated previously, by other methods, in L.I. Ignat and J.D. Rossi (2008) [11]